By Theorem 8.2 [Weak Law of Large Numbers] (pg. 54-55) of
[1]}, \(\hat{g}_{*,n} \overset{\mathcal {P}}{\rightarrow }g_*\) . Further, if \(G\)  is uniform over the discrete set of points \([\Delta +1,\ldots , \Delta +M]\) , then for \(y \in [\Delta + 1, \ldots , \Delta + M]\)
\(\beta (y) = \frac{\Pr (Y = y)}{\Pr (Y \le y)} = \frac{1}{M} \frac{M}{y - (\Delta + 1)+1} = \frac{1}{y - \Delta }.\) 
